The Stability of Stationary Fronts for a Discrete Nagumo Equation
C.E. Elmer, Mathematical Biosciences and Engineering 4:1 (2007) pp. 113–129.
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Abstract: We consider the stability of single front stationary solutions to a spatially discrete reaction-diffusion equation which models front propagation in a nerve axon. The solution’s stability depends on the coupling parameter, changing from stable to unstable and from unstable to stable at a countably infinite number of values of this diffusion coefficient.

Finding Stationary Fronts for a Discrete Nagumo and Wave Equation, Construction
C.E. Elmer, Physica D 218:1 (2006) pp. 11–23.
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Abstract: We present single front stationary solutions for a spatially discrete reaction-diffusion equation, and a wave equation, with a piecewise linear zigzag shaped nonlinearity. Of notable interest is that a region of propagation failure does not exist for an infinite number of isolated values of the diffusion parameter.

Spatially Discrete Fitz-Hugh Nagumo Equations
C.E. Elmer and E.S. Van Vleck, SIAM Journal on Applied Mathematics 65, 4 (2005) pp. 1153-117.
 Paper (.pdf)
Abstract: We consider pulse and front solutions to a spatially discrete FitzHugh-Nagumo equation that contains terms to represent both depolarization and hyperpolarization of the nerve axon. We demonstrate a technique for deriving candidate solutions for the McKean nonlinearity and present and apply solvability conditions necessary for existence. Our equation contains both spatially continuous and discrete diffusion terms.

Dynamics of Monotone Traveling Fronts for Discretizations of Nagumo PDE's
C.E. Elmer and E.S. Van Vleck,  Nonlinearity 18, 4 (2005) pp. 1605-1628. 
Paper (.pdf),
Abstract: When PDEs are discretized, the dynamics of the resulting equations differ from those of the original PDE. In this paper, we study the dynamics of travelling wave solutions to the discretized Nagumo PDE (A(α) stable in time and/or uniform in space) with smooth bistable nonlinearities. In general, time discretization significantly speeds up travelling wave fronts and spatial discretization slows (or even halts) such fronts.

Computation of Mixed Type Functional Differential Boundary Value Problems
K.A. Abell, C.E. Elmer, A.R. Humphries and E.S. Van Vleck,  SIAM Journal on Applied Dynamical Systems 4:3 (2005) 755-781.
Paper (.pdf)
Abstract: We study boundary value differential-difference equations where the difference terms may contain both advances and delays. Special attention is paid to connecting orbits, in particular to the modeling of the tails after truncation to a finite interval, and we reformulate these problems as functional differential equations over a bounded domain. Connecting orbits are computed for several such problems including discrete Nagumo equations, an Ising model, and Frenkel--Kontorova type equations. We describe the collocation boundary value problem code used to compute these solutions, and the numerical analysis issues which arise, including linear algebra, boundary functions and conditions, and convergence theory for the collocation approximation on finite intervals.


Anisotropy, Propagation Failure, and Wave Speedup in Traveling Waves of Discretizations of a Nagumo PDE
C.E. Elmer and E.S. Van Vleck, Journal of Computational Physics 185, 2 (2003) pp. 562–-582. 
Paper (.pdf)
Abstract: This article is concerned with effect of spatial and temporal discretizations on traveling wave solutions to parabolic PDEs (Nagumo type) possessing piecewise linear bistable nonlinearities. Solution behavior is compared in terms of waveforms and in terms of the so-called (a,c) relationship where a is a parameter controlling the bistable nonlinearity by varying the potential energy difference of the two phases and c is the wave speed of the traveling wave. Uniform spatial discretizations and A-alpha stable linear multistep methods in time are considered. Results obtained show that although the traveling wave solutions to parabolic PDEs are stationary for only one value of the parameter a; a0, spatial discretization of these PDEs produce traveling waves which are stationary for a nontrivial interval of a values which include a0, i.e., failure of the solution to propagate in the presence of a driving force. This is true no matter how wide the interface is with respect to the discretization. For temporal discretizations at large wave speeds the set of parameter a values for which there are traveling wave solutions is constrained. An analysis of a complete discretization points out the potential for nonuniqueness in the (a,c) relationship.


Existence of Monotone Traveling Fronts for BDF Discretizations of Bistable Reaction-Diffusion Equations
C.E. Elmer and E.S. Van Vleck, Journal of Dynamics of Continuous, Discrete and Impulsive Systems 10A, pp. 389–-402.
Paper (.pdf)
Abstract: This article is concerned with the effect of temporal discretization on traveling wave solutions to parabolic PDEs possessing bistable nonlinearities. The focus is on the application of backward differentiation formulas to Nagumo type PDEs with two different bistable nonlinearities. Existence of monotone traveling fronts is shown and the efficacy of different methods of proof is discussed.

A Variant of Newton's Method for the Solution of Traveling Wave Solutions of Bistable Differential-Difference Equations
C.E. Elmer and E.S. Van Vleck,  J. Dyn. Diff. Eqns. 14, 3 (2002) 493-517. 
Paper (.pdf)
Abstract: We consider a variant of Newton’s method for solving nonlinear differential- difference equations arising from the traveling wave equations of a large class of nonlinear evolution equations. Building on the Fredholm theory recently developed by Mallet-Paret we prove convergence of the method. The utility of the method is demonstrated with a series of examples.

Traveling Wave Solutions for Bistable Differential-Difference Equations with Periodic Diffusion
C.E. Elmer and E.S. Van Vleck, SIAM J. Appld. Math. 61, 5 (2001) 1648-1679. 
Paper (.pdf)
Abstract: We consider traveling wave solutions to spatially discrete reaction-diffusion equations with nonlocal variable diffusion and bistable nonlinearities. To find the traveling wave solutions we introduce an ansatz in which the wave speed depends on the underlying lattice as well as on time. For the case of spatially periodic diffusion we obtain analytic solutions for the traveling wave problem using a piecewise linear nonlinearity. The formula for the wave forms is implicitly defined in the general periodic case and we provide an explicit formula for the case of period two diffusion. We present numerical studies for time t = 0 fixed and for the time evolution of the traveling waves. When t = 0 we study the cases of homogeneous, period two, and period four diffusion coefficients using a cubic nonlinearity, and uncover, numerically, a period doubling bifurcation in the wave speed versus detuning parameter relation. For the time evolution case we also discover a detuning parameter dependent bifurcation in observed phenomena, which is a product of both the nonlocal diffusion operator and the spinodal effects of the nonlinearity.

Analysis and Computation of Traveling Wave Solutions of Bistable Differential-Difference Equations
C.E. Elmer and E.S. Van Vleck, Nonlinearity 12, 4 (1999) pgs. 771-798. 
Paper (.pdf)
 Abstract: We consider travelling wave solutions of a class of differential-difference equations. Our interest is in understanding propagation failure, the directional dependence due to the discrete Laplacian, and the relationship between travelling wave solutions of the spatially continuous and spatially discrete limits of this equation. The differential-difference equations that we study include damped and undamped nonlinear wave and reaction-diffusion equations as well as their spatially discrete counterparts. Both analytical and numerical results are given.

Computation of Traveling Wave Solutions for Spatially Discrete Bistable Reaction-Diffusion Equations
C.E. Elmer and E.S. Van Vleck,  Appld. Numer. Math. 20, 1-2 (1996) 157--169.
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Abstract: Traveling wave solutions for reaction-diffusion equations on a discrete spatial domain are considered. Traveling wave equations are derived for the spatial domain Z^n for n=1,2,3. Using an idealized nonlinear term, the anisotropy introduced by the lattice is analyzed. Numerical techniques for solving the traveling wave equations are introduced. Finally, some numerical experiments are presented.

A Construction Technique for Heteroclinic Solutions to Continuous and Differential-Difference Damped Wave Equations
M. Rodrigo, C.E. Elmer, and R.M. Miura, CAMS Technical Report 0203-24, NJIT (2003).  
Paper (.pdf)
Abstract: In this paper, we give a systematic method for generating continuous and differential-difference damped wave equations for which explicit travelling wave solutions can be obtained. We demonstrate the procedure with several examples. In some specific cases, we recover the well-known solutions of the continuous Nagumo and sine-Gordon equations.


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